How many prime numbers are there between 1,000,000 and 1,100,000?

Question

mathmate · Answer

Does that have to do with log(n)/n and the Riemann-Zeta function?

mathmate · Answer

Oops, n/log(n)

across · Answer

With this question, I am seeking to clash a mathematical approach with a computerized one. Although it would be interesting to see in how many different ways we can tackle this exercise.

mathmate · Answer

So that makes 6692.7, say 6693.
Is there a better estimate using R-Z function, I don't really understand it.

across · Answer

mathmate, your approximation is fairly close!

across · Answer

For corroboration, I wrote this: http://ideone.com/qoDbe I am now trying an analytical approach.

mathmate · Answer

Can you elaborate on your analytical approach, or is it proprietary?

mathmate · Answer

6693 wasn't even close, you were being polite!
Using the "offset logarithmic integral"
$$Li(n) = \int\limits_{2}^{n}\ \frac{dt}{\log(t)}$$
I get 7212.99 > 7213.
Using your code, I get 7216.
So now it's getting close.

Also, your code included 1 as a prime, which it is not.  This does not change the counts over 1 though.

across · Answer

That made me want to go back and recheck the code. ^^

anonymous · Answer

wow.....

across · Answer

Interesting. Using the logarithmic integral from 2,000,000 to 2,100,000 you get 6881, whereas the (fixed) code chunks out 6871.

The error is now negative.

anonymous · Answer

Code looks good except comparison to 1.

anonymous · Answer

btw - why picture change?

mathmate · Answer

I get the same answers, except that I get 6872 for the actual count, both using your code and my code.